Abstract

We improve the class of subcompatible self-maps used by (Akbar and Khan, 2009) by introducing a new class of noncommuting self-maps called modified subcompatible self-maps. For this new class, we establish some common fixed point results and obtain several invariant approximation results as applications. In support of the proved results, we also furnish some illustrative examples.

1. Introduction and Preliminaries

From the last five decades, fixed point theorems have been used in many instances in invariant approximation theory. The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1] where he employs a fixed point theorem of Schauder to establish the existence of an invariant approximation. Later on, Brosowski [2] used fixed point theory to establish some interesting results on invariant approximation in the setting of normed spaces and generalized Meinardus’s results. Singh [3], Habiniak [4], Sahab et al. [5], and Jungck and Sessa [6] proved some similar results in the best approximation theory. Further, Al-Thagafi [7] extended these works and proved some invariant approximation results for commuting self-maps. Al-Thagafi results have been further extended by Hussain and Jungck [8], Shahzad [9–14] and O’Regan and Shahzad [15] to various class of noncommuting self-maps, in particular to R-subweakly commuting and R-subcommuting self-maps. Recently, Akbar and Khan [16] extended the work of [7–15] to more general noncommuting class, namely, the class of subcompatible self-maps.

In this paper, we improve the class of subcompatible self-maps used by Akbar and Khan [16] by introducing a new class of noncommuting self-maps called modified subcompatible self-maps which contain commuting, R-subcommuting, R-subweakly, commuting, and subcompatible maps as a proper subclass. For this new class, we establish some common fixed point results for some families of self-maps and obtain several invariant approximation results as applications. The proved results improve and extend the corresponding results of [3–8, 10–15].

Before going to the main work, we need some preliminaries which are as follows.

Definition 1. Let be a metric space, be a subset of , and and be self-maps of . Then the family of self-maps of is called :(i)contraction if there exists , such that for all ,
(ii)nonexpansive if for all ,
In Definition 1, if we take , then this family is called -contraction (resp., -nonexpansive).

Definition 2. Let be a subset of a metric space and be self-maps of . A point is a coincidence point (common fixed point) of and if . The set of coincidence points of and is denoted by . The pair is called(1)commuting if for all ;(2) R-weakly commuting [17], provided there exists some positive real number R such that for each ;(3)compatible [18] if = 0 whenever is a sequence in such that for some ;(4)weakly compatible [19] if for all .For a useful discussion on these classes, that is, the class of commuting, R-weakly commuting, compatible, and weakly compatible maps, see also [20].

Definition 3. Let be a linear space and let be a subset of . The set is said to be star-shaped if there exists at least one point such that the line segment joining to is contained in for all ; that is, for all , where .

Definition 4. Let be a linear space and let be a subset of . A self-map is said to be(i)affine [21] if is convex and
(ii)q-affine [21] if is -star-shaped and
Here we observe that if is -affine then .

Remark 5. Every affine map is -affine if but its converse need not be true even if , as shown by the following examples.

Example 6. Let and . Let be defined as
Then is -affine for , while is not affine because for , , and
does not hold.

Example 7. Let and . Let , where
Then is -star-shaped for . Define as
Then is -affine for but is not affine, because for , , and , , though .

Definition 8. Let be a subset of a normed linear space . The set is called the set of best approximants to out of , where .

Definition 9 (see [11]). Let be a subset of a normed linear space and let and be self-maps of . Then the pair is called -subweakly commuting on with respect to if is -star-shaped with (where denote the set of fixed point of ) and for all and some .

Definition 10. Let be a Banach space. A map is said to be demiclosed at 0 whenever is a sequence in such that converges weakly to and converges strongly to ; then .

Definition 11. A Banach space is said to satisfy Opial’s condition whenever is a sequence in such that converges weakly to ; then
Note that Hilbert and spaces satisfy Opial’s condition.

2. Common Fixed Point for Modified Subcompatible Self-Maps

First we introduce the notion of modified subcompatible maps.

Definition 12. Let be a -star-shaped subset of a normed linear space and let and be self-maps of with . Define , where and . Then and are called modified subcompatible if = 0 for all sequences .

In the definition of subcompatible maps (see [16]), , but here . The following examples reveal the impact of this and show that -subweakly commuting maps and also subcompatible maps of [16] form a proper subclass of modified subcompatible maps.

Example 13. Let with the usual norm and . Define by
Then is 1-star-shaped with and . Moreover, and are modified subcompatible but not subcompatible because for the sequence , we have and Note that and are neither R-subweakly commuting nor R-subcommuting.

Example 14. Let with the usual norm and . Define by
Then is -star-shaped with and . Clearly and are modified subcompatible but not subcompatible because for any sequence , we have and Also, and are not R-subweakly commuting.

The following two examples show that the modified subcompatible self-maps and compatible self-maps are of different classes.

Example 15. Let with usual norm and . Let be defined by
for all . Then
Thus and are compatible. Obviously is -star-shaped with and . Note that for any sequence in with , we have
However, . Thus and are not modified subcompatible maps. Hence, they are not -subweakly commuting.

Example 16. Let with norm and . Let be defined by
Then is 3-star-shaped with and . Clearly and are modified subcompatible. Moreover, for any sequence in with , we have . However, . Thus and are not compatible.

The following general common fixed point result is a consequence of Theorem 5.1 of Jachymski [22], which will be needed in the sequel.

Theorem 17. Let and be self-maps of a complete metric space and either or is continuous. Suppose is a sequence of self-maps of satisfying the following.(1) and for each .(2)The pairs and are compatible for each .(3)For each and, for any ,
where
then there exists a unique point in such that , for each .

Theorem 18. Let be a nonempty -star-shaped subset of a normed space and let and be continuous and -affine self-maps of . Let be a family of self-maps of satisfying the following.(1) and for each .(2) and are modified subcompatible for each .(3)For each and, for any where
then all the , and have a common fixed point provided one of the following conditions hold.(a) is sequentially compact and is continuous for each .(b) is weakly compact, is demiclosed at 0 for each , and is complete.

Proof. For each , define by
for all and a fixed sequence of real numbers converging to 1. Then, is a self-map of for each and for each.Firstly, we prove () ; for this let (), which implies for some .Now, by using (20)
Hence for each .Similarly, it can be shown that for each and each , , as is -affine and is -star-shaped.Now, we prove that for each , the pair is compatible; for this let with . Since the pair () is modified subcompatible, therefore, by the assumption of , we have
As the pair is modified subcompatible and is -affine, therefore
Hence, the pair is compatible for each .Similarly, we can prove that the pair () is compatible for each and each .Also, using (18) and (20) we have
for each and . By Theorem 17, for each , there exists such that , for each .(a)As is sequentially compact and is a sequence in , so has a convergent subsequence such that . Thus, by the continuity of , and all (), we can say that is a common fixed point of , and all . Thus .(b)Since is weakly compact, there is a subsequence of converging weakly to some . But, and being -affine and continuous are weakly continuous, and the weak topology is Hausdorff, so is a common fixed point of and . Again the set is bounded, so = 0 as . Now demiclosedness of at 0 gives that for each , and hence .

Theorem 19. Let be a nonempty -star-shaped subset of a normed space , and let and be continuous and -affine self-maps of . Let be a family of self-maps with and for each . If the pairs and are modified subcompatible for each and also the family of maps is -nonexpansive, then , provided one of the following conditions hold.(a) is sequentially compact.(b) is weakly compact, is demiclosed at 0 for each , and is complete.(c) is weakly compact and is a complete space satisfying Opial’s condition.

Proof. (a) The proof follows from Theorem 18(a).(b) The proof follows from Theorem 18(b).(c) Following the proof of Theorem 18(b), we have and for each , as Since the family is -nonexpansive, therefore, for each , we have . Now we have to show that . If not, then by Opial’s condition of and ()-nonexpansiveness of the family , we get
which is a contradiction. Therefore, and, hence, .

In Theorems 18 and 19, if we take for each , we obtain the following corollary which generalizes Theorems 2.2 and 2.3 of Hussain and Jungck [8], respectively.

Corollary 20. Let be a nonempty q-star-shaped subset of a normed space , and let and be continuous and -affine self-maps of . Let be a self-map of satisfying the following.(1).(2)The pairs and are modified subcompatible.(3)For all ,
where
Then S, T, and have a common fixed point provided one of the following conditions hold.(a) is sequentially compact and is continuous.(b) is weakly compact, is demiclosed at 0, and is complete.(c) is complete, is compact, and is continuous.

Proof. (a) and (b) follow from Theorem 18 by taking for each .(c) Define : by
As we have done in Theorem 18, for each , there exists such that . Then, compactness of cl implies that there exists a subsequence of such that as. Then the definition of implies ; thus, by continuity of , and , we can say that is a common fixed point of , and .

Corollary 21. Let be a nonempty -star-shaped subset of a normed space , and let and be continuous and -affine self-maps of . Let be a self-map of with . If the pairs and (,) are modified subcompatible and also the map is -nonexpansive, then , provided one of the following conditions hold.(a) is sequentially compact.(b) is weakly compact, is demiclosed at 0, and is complete.(c) is weakly compact and is complete space satisfying Opial’s condition.(d) is complete and is compact.

In Corollary 20(b), if we take , then we obtain the following corollary as a generalization of Theorem 4 proved by Shahzad [12].

Corollary 22. Let be a nonempty weakly compact -star-shaped subset of a Banach space , and let and be self-maps of . Suppose that is -affine and continuous, and . If is demiclosed at 0, the pair is modified subcompatible and satisfies
where
for all ; then .

In Theorems 18 and 19, if we take , then we obtain the following corollary.

Corollary 23. Let be a nonempty -star-shaped subset of a normed space . Suppose that is continuous and is a -affine self-map of . Let be a family of self-maps of satisfying the following.(1) and for each , the pair is modified subcompatible.(2)For each and, for any
where
then and all the have a common fixed point provided one of the following conditions hold.(a) is sequentially compact and is continuous for each .(b) is weakly compact, is demiclosed at 0 for each , and is complete.

Corollary 24. Let be a nonempty -star-shaped subset of a normed space . Suppose that is continuous and is a -affine self-map of . Let be a family of self-maps with and the pairs are modified subcompatible for each . If this family of maps is -nonexpansive then , provided one of the following conditions hold.(1) is sequentially compact.(2) is weakly compact, is demiclosed at 0 for each , and is complete.(3) is weakly compact and is a complete space satisfying Opial’s condition.

3. Applications to Best Approximation

The following theorem extends and generalizes [5, Theorem 2], [8, Theorem 2.8], and main result of [3].

Theorem 25. Let be a subset of a normed space and let , , be mappings for each such that for some and for each , . Suppose that and are -affine and continuous on and also is -star-shaped and .Moreover, if (1)the pairs and are modified subcompatible for each .(2)for each , and for all ,
Then , provided one of the following conditions hold.(a) is sequentially compact and is continuous for each .(b) is weakly compact, is complete, and () is demiclosed at 0 for each .

Proof. Let . Then . Note that for any ,
It follows that the line segment and the set are disjoint. Thus, is not interior of and so As for each , therefore, for each , Now we have to show that and for each , . Since , and , and ’s satisfy (33); therefore, we have
Then the definition of implies
Again using (33) and (34), for each , we have
This yields that
Then combining and , we get for each . Consequently, , for each . Since , therefore we have
Hence, by Theorem 18.

Corollary 26. Let be a subset of a normed space and let , , : → be mappings for each such that for some and for each . Suppose that and are -affine and continuous on and also is q-star-shaped and . If the pairs and are modified subcompatible for each and also the family of maps is -nonexpansive, then , provided one of the following conditions hold.(a) is sequentially compact.(b) is weakly compact, is complete, and is demiclosed at 0 for each .(c)is weakly compact and is complete space satisfying Opial’s condition.

Corollary 27. Let be a subset of a normed space and let , be mappings such that for some and . Suppose that is -affine and continuous on and also is -star-shaped and . If the pair is modified subcompatible and satisfies for all
then provided one of the following conditions hold.(a) is sequentially compact.(b) is complete and is compact.(c) is weakly compact, is complete, and is demiclosed at 0.

4. Examples

Now, we present some examples which demonstrate the validity of the proved results.

Example 28. Let with usual norm = and . Suppose , are defined as
and also are defined as
Here , , and for each , so that and for each . Besides is compact and the pairs of mappings and are modified subcompatible for each and also the maps and are -affine for . Further the mappings , and for each satisfy the inequality (18). Hence all the conditions of Theorem 18(a) are satisfied. Therefore , and all have a common fixed point and is such a unique common fixed point.

Remark 29. (1) In Example 28, if we define for all , then , , and all are self-maps of and . Clearly, is -star-shaped and . Therefore, all the conditions of Theorem 25 are satisfied and, hence, . Here, .(2) If inequality (18) in Theorem 18 is replaced with the weaker condition
for each and, for any . Then, Theorem 18 need not be true. This can be seen by the following example.

Example 30. Let with usual norm = and . Suppose , are defined as
and also are defined as
Here , , and for each , so that and for each . Besides is compact and the pairs of mappings and are modified subcompatible for each and also the maps S and T are -affine for . Further, the mappings , , and for each are continuous and satisfy the inequality (42). Note that .

Remark 31. Clearly mappings , and for each defined in Example 30 satisfy all of the conditions of Theorem 18(a) except the inequality (18) at , . Note that there is no common fixed point of , and for each .

Example 32. Let with usual norm = and . Suppose are defined as
Here we observe that , , and so that . Also, is -star-shaped and the maps and are -affine with . We also observe that the pairs and are modified subcompatible and is sequentially compact. Further, the mappings , , and satisfy (26). Hence, the mappings ,, and satisfy all the conditions of Corollary 20(a) and is the unique common fixed point of mappings , , and .

Remark 33. In Example 32, and are not affine because for , , and , and do not hold. Therefore, Theorem 2.2 of Hussain and Jungck [8] cannot apply to Example 32; hence Corollary 20 is more general than Theorem 2.2 of [8].

Example 34. Take , , and as in Example 32 and define
Then all of the conditions of Corollary 20(a) are satisfied except that the pair is modified subcompatible. Note that .

Remark 35. All results of the paper can be proved for Hausdorff locally convex spaces defined and studied by various authors (see [16, 23–27]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.